Electronic – BCR402R – what is the purpose of Rext in this LED driver

There are various misconceptions here.

The emitter resistor value has no effect on efficiency. All the current for each LED is coming from the 5V supply. Whatever part of that voltage the LED doesn't use times the current is going to be wasted as heat. Small emitter resistors only more the dissipation to the transistors. The total dissipation is the same.

I thought I mentioned it in your other question, but the emitter resistors should be a larger value. At 1 Ohm just 1 1mV offset will cause 1mA thru the LED, which is probably dim but visible. From your voltage divider ratio and the 1 Ohm emitter resistors, it looks like you're aiming at a bit over 300mA LED current. Didn't I go thru a detailed calculation of the emitter resistor value in your other question?

I don't know what kind of LEDs these are, but most likely you can afford at least a volt accross the emitter resistor, so 3.3 Ohms would be a better choice that will give you more control.

As for the not quite 0 and 5 volt output, that's probably something the Arduino is doing. Keep in mind that arduinos are Simplified for the masses. It wouldn't surprise me if there is a resistor in series with each output as protection.

The opamps are probably working fine enough. Every opamp has some offset error, and if these are a little positive, their outputs will go high just enough to turn the transistors on a little to make that offset voltage appear accross the emitter resistor. This is yet another reason for using a larger emitter resistor.

To understand how these devices function, it helps to understand how traditional LEDs function. An LED is a Light Emitting Diode, so basically you can think of it as a simple PN junction that would be used in a diode (although in actuality the structure of LEDs is more complicated, often a double heterojunction structure).

_{(source: gsu.edu)}

Based on images of a similar LED "filament" bulb on Amazon image1, image2

It seems that each "filament" is a single LED with a radial structure, that is using radial layers rather than the traditional planar layers. So the core layer (cathode for example) would be composed of a conductor/metal, then the next outer layer would be n-type material, then the next outer layer would be p-type material, and finally the outermost layer would be a transparent conductor (to let light pass through it) such as ITO (Indium-Tin-Oxide).

It looks as though the LEDs in the bulb are arranged as two lines in parallel, each with two LEDs in series.

The driver would depend on the specs of the particular LED "filaments", but a driver could be as simple as diode bridge if the LEDs have a high enough forward voltage.

Because these LED "filaments" are so long and thin, they have a large amount of surface-area per volume, so that the heat is more dissipated than in a traditional LED.